Geodesic flow for $\mathrm{CAT}(0)$–groups

Abstract

We associate to a $CAT\left(0\right)$–space a flow space that can be used as the replacement for the geodesic flow on the sphere tangent bundle of a Riemannian manifold. We use this flow space to prove that $CAT\left(0\right)$–group are transfer reducible over the family of virtually cyclic groups. This result is an important ingredient in our proof of the Farrell–Jones Conjecture for these groups.